Heat Transfer Through a Plane Wall

In summary, the conversation discusses solving a heat transfer problem involving a large plane wall with constant thermal conductivity and no heat generation. The solution involves solving a differential equation and applying boundary conditions to determine the temperature distribution in the wall. The rate of heat transfer through the wall is evaluated to be 7079 W, which is different from the answer provided in the book. The conversation also mentions the use of thermal resistance and the interchangeability of temperature units. The book in question is Heat and Mass Transfer Fundamentals & Applications 4th Ed by Cengel and Ghajar.
  • #1
bob1182006
492
1

Homework Statement


Consider a large plane wall of thickness L = 0.4m, thermal conductivity k = 1.8W/(m*K), and surface area A = 30m2. The left side of the wall is maintained at a constant temperature of T1 = 90 C while the right side looses heat by convection to the surrounding air at Ts = 25 C with a heat transfer coefficient of h = 24 W(m2*K). Assuming constant thermal conductivity and no heat generation in the wall evaluate the rate of heat transfer through the wall.
Answer: 7079 W

My answer doesn't match the book's answer.

Homework Equations


[tex]\dot{Q}_{wall} = -kA\frac{dT(0)}{dx}[/tex]
[tex]\frac{d^2T}{dx^2} = 0[/tex]
[tex]T(0) = 90[/tex]
[tex]-k\frac{dT(L)}{dx} = h[T(L) - Ts][/tex]

The Attempt at a Solution



Solving the differential equation and applying B.C.:
[tex]T(x) = xC_1 + C_2[/tex]
[tex]T(0) = C_2 = 90[/tex]
[tex]-kC_1 = hLC_1+hC_2-25h][/tex]
[tex]C_1 = -\frac{h(C_2-25)}{k+hL}[/tex]

Plugging numbers in:
[tex]C_2 = 90[/tex]
[tex]C_1 = -136.8[/tex]
[tex]T(x) = 90-136.8x[/tex]
[tex]\dot{Q}_{wall} = -1.8*30*(-136.8) = 7,387 W[/tex]

Did I make a mistake or is the book's answer wrong?
 
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  • #2
I used a somewhat different approach (underlying equations are of course the same) and got the same answer you did.

I computed the thermal resitance of the wall as L/kA = 7.41e-3 K/W and the equivalent thermal resistance of the convection effect as 1/hA = 1.39e-3 K/W, giving total thermal resistance of 8.80e-3 K/W so dQ/dt = (90 - 25)/8.80e-3 = 7387W.

Just curious - what is your textbook?
 
  • #3
Awesome thanks, I didn't want to waste any more time redoing this problem looking for a mistake, and my class is just starting the thermal resistance chapter so if I see more problems that have wrong answers I'll do them that way to check.

We're using Heat and Mass Transfer Fundamentals & Applications 4th Ed by Cengel and Ghajar.
 
  • #4
arent we suppose to take temp. as kelvin here?
 
  • #5
spree said:
arent we suppose to take temp. as kelvin here?

It's OK to interchange K and C as long as you're dealing with temperature differences.
 
  • #6
oh ure right. new to the subject, just trying to learn sorry :) btw I personally know professor cengel and i will inform him about this. I will let you know about it. if you suspect any other mistake in the book feel free to ask please.
 

What is heat transfer through a plane wall?

Heat transfer through a plane wall is the process by which thermal energy is transferred from one side of a flat surface (the plane wall) to the other side. This can occur through three main methods: conduction, convection, and radiation.

What is conduction?

Conduction is the transfer of heat through a solid material or stationary fluid, such as air. In the context of heat transfer through a plane wall, it refers to the transfer of heat from one side of the wall to the other through direct contact between the molecules of the material.

What is convection?

Convection is the transfer of heat through a fluid, such as air or water, due to the movement of the fluid. In the case of heat transfer through a plane wall, convection can occur if there is a temperature difference between the two sides of the wall, causing the air or fluid to move and transfer heat.

What is radiation?

Radiation is the transfer of heat through electromagnetic waves. In the context of heat transfer through a plane wall, it refers to the transfer of heat through infrared radiation between the two sides of the wall.

What factors affect heat transfer through a plane wall?

Several factors can affect heat transfer through a plane wall, including the thickness and material of the wall, the temperature difference between the two sides, and the presence of any insulating materials. Additionally, the area of the wall, the type of surface (rough or smooth), and the direction of heat flow can also impact the rate of heat transfer.

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